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Order in the Chaotic region

  • Eduardo Piña
Workshop
Part of the Lecture Notes in Physics book series (LNP, volume 189)

Abstract

A simple algorithm is constructed for the quadratic one-parameter family of maps. It predicts the number of periodic orbits of arbitrary period ocurring for parameter values lower than any other corresponding to a given stable periodic orbit. This algorithm produces the number of periodic unstable points coexisting with the stable periodic orbit. This method associates an important polynomial in a one-to-one correspondence with the symbol of Metropolis et al. for ordering the periodic orbits, with the permutation matrix of the dynamics of points in the stable cycle, and with the matrix of regions separated by these points.

These polynomials coincide with those polynomials used by many authors in connection with the triangular map and with the topological entropy.

Keywords

Periodic Orbit Characteristic Polynomial Periodic Point Topological Entropy Large Root 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    A. Libchaber and J. Maurer, Local probe in a Rayleigh-Benard experiment in liquid helium. J. Phys. (Paris) Lett. 39 L, 369–372; Yu N. Belyaev, A. A. Monakhov, S. A. Sherbakov, and I. M. Yavorskaya, Onset of turbulence in rotating fluids. JETP Lett. 29, 295–298 (1979); J. P. Gollub, S. V. Benson, and J. Steinman, A subharmonic route to turbulent convection. Ann. N.Y. Acad. Sci. 357, 22–27 (1981); M. Giglio, S. Musazzi, and U. Perini, Transition to chaotic behavior via a reproducible sequence of period-doubling bifurcations, Phys. Rev. Lett. 47, 243–246 (1981).Google Scholar
  2. [2]
    R. H. Simoyi, A. Wolf and H. L. Swinney, One-dimensional dynamics in a multicomponent chemical reaction, Phys. Rev. Lett. 49, 245–248 (1982).CrossRefGoogle Scholar
  3. [3]
    J. Testa, J. Perez, and C. Jeffries, Evidences for universal chaotic behavior of a driven nonlinear oscillator, Phys. Rev. Lett. 48, 714–717 (1982).CrossRefGoogle Scholar
  4. [4]
    P. S. Linsay, Period doubling and chaotic behavior in a driven enharmonic oscillator, Phys. Rev. Lett. 47, 1349–1352 (1981); F. T. Arecchi and F. Lisi, Hopping mechanism generating 1/f noise in nonlinear systems. Phys. Rev. Lett. 49, 94–98 (1982); R. W. Rollins and E. R. Hunt, Exactly solvable model of a physical system exhibiting universal chaotic behavior. Phys. Rev. Lett. 49, 1295–1298 (1982).CrossRefGoogle Scholar
  5. [5]
    H. M. Gibbs, F. A. Hopf, D. L. Kaplan, and R. L. Shoemaker, Observation of chaos in optical bistability. Phys. Rev. Lett. 46, 474–477 (1981); F. T. Arecci, R. Meucci, G. Puccioni, and J. Tradicce, Experimental evidence of subharmonic bifurcations, multistability and turbulence in a Q-switched gas laser. Phys. Rev. Lett. 49, 1217–1220 (1982).CrossRefGoogle Scholar
  6. [6]
    R. Keolian, L. A. Turkevich, S. J. Putterman, I. Rudnick, and J. A. Rudnick, Subharmonic sequences in the Faraday experiment: departures from period doubling. Phys. Rev. Lett. 47, 1133–1136 (1981); W. Lauterborn and E. Cramer, Subharmonic route to chaos observed in acoustics. Phys. Rev. Lett. 47, 1445–1448 (1981); C. W. Smith, M. J. Tejwani and D. A. Farris, Bifurcation universality for first-sound subharmonic generation in superfluid helium 4. Phys. Rev. Lett. 48, 492–494 (1982).CrossRefGoogle Scholar
  7. [7]
    R. M. May, Simple mathematical models with very complicated dynamics. Nature 281, 459–467 (1976); P. J. Myrberg, Iteration von Quadratwurzeloperationen, Iteration der Reelen Polynome Zweiten Grades. Ann. Akad. Sci. Fennicæ A, I Nos. 259, (1958), 336/3, (1963); E. N. Lorenz, The problem of deducing the climate from the governing equations. Tellus 16, 1–11 (1964); P. Collet and J. P. Eckman, Iterated maps on the interval as dynamcal systems. Birkhäuser, Basel, 1980.Google Scholar
  8. [8]
    N. Metropolis, M. L. Stein, and P. R. Stein, On finite limit sets for transformations on the unit interval. J. Combinatorial Theory 15A, 25–44 (1973).CrossRefGoogle Scholar
  9. [9]
    M. J. Feigenbaum, Quantitative universality for a class of nonlinear transformations. J. of Stat. Phys. 19, 25–52 (1978); ib., The universal metric properties of nonlinear transformations. J. of Stat. Phys. 21, 669–706 (1979); ib., The transition to aperiodic behavior in turbulent systems. Commun. Mat. Phys. 77, 65–68 (1980).CrossRefGoogle Scholar
  10. [10]
    B. Derrida, A. Gervois, and Y. Pomeau, Iteration of endomorphisms on the real axis and representation of numbers. Ann. Inst. Henri Poincaré A29, 305–356 (1978).Google Scholar
  11. [11]
    J. Guckenheimer, On the bifurcation of maps of the interval. Invent. Math. 39, 165–178 (1977).CrossRefGoogle Scholar
  12. [12]
    O. Stefan, A theorem of Sarkovskiî on the existence of periodic orbits of continuous endomorphisms of the real line. Commun. Math. Phys. 54, 237–248 (1977).Google Scholar
  13. [13]
    S. Smale and R. Williams, The quantitative analysis of a difference equation of population growth. J. Math. Biol. 3, 1–4 (1976).PubMedGoogle Scholar
  14. [14]
    J. Guckenheimer, G. Oster, and A. Ipaktchi, The dynamics of density dependent population models. J. Math. Biol. 4, 101–147 (1977).Google Scholar
  15. [15]
    C. Jordan, Calculus ofFinite Differences, Chelsea Publishing Co., 1979. p. 593.Google Scholar
  16. [16]
    R. S. Varga, Matrix Iterative Analysis. Prentice-Hall, 1962.Google Scholar
  17. [17]
    J. Guckenheimer, Sensitive dependence to initial conditions for one dimensional maps. Commun. Math. Phys. 70, 133–160 (1979).Google Scholar
  18. [18]
    L. Block, J. Guckenheimer, M. Misiurewicz, and L. S. Young, Periodic Points and Topological Entropy of One Dimensional Maps. Lecture Notes in Mathematics, #819, Spriger Verlag, 1980. pp. 18–34.Google Scholar
  19. [19]
    D. Ruelle, Applications consevant une measure absolument continue por rapport a dx sur [0,1]. Commun. Math. Phys. 55, 47 (1977).Google Scholar
  20. [20]
    F. C. Hoppenstead and J. M. Hyman, Periodic solutions of a logistic difference equation. SIAM J. Appl. Math. 32, 73–81 (1977).Google Scholar
  21. [21]
    S. Grossman and S. Thomae, Invariant distributions and stationary correlation functions of one-dimensional discrete processes. Z. Naturforsch. 32a, 1353–1363 (1977).Google Scholar
  22. [22]
    S. J. Chang and J. Wright, Transitions and distribution functions for chaotic systems. Phys. Rev. A23, 1419–1433 (1981).Google Scholar
  23. [23]
    R. Shaw, Stange attractors, chaotic behavior, and information flow. Z. Naturforsch. 36a, 80–112 (1981.)Google Scholar
  24. [24]
    J. Dias de Deus, R. Dilão, and J. Taborda Duarte, Topological entropy and approaches to chaos in dynamics of the interval. Phys. Lett. 90A, 1–4 (1982).Google Scholar
  25. [25]
    K. Goldberg, M. Newman, and E. Haynsworth, in Handbook of Mathematical Functions. (M. Abramowitz and I. A. Stegun eds.) Dover Publ. 1965. p. 826.Google Scholar
  26. [26]
    E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences. Illinois J. Math. 5, 657–665 (1961).Google Scholar
  27. [27]
    O. Chavoya, F. Angulo, and E. Piña. (To be published.)Google Scholar

Copyright information

© Springer-Verlag 1983

Authors and Affiliations

  • Eduardo Piña
    • 1
    • 2
  1. 1.Departamento de FísicaUniversidad Autónoma MetropolitanaIztapalapa D. F.Mexico
  2. 2.Escuela Superior de Física y MatemáticasInstituto Politécnico NacionalZacatenco, D.F.México

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